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Energy Minimization
Aishwarya M. Rane
M.Sc. Bioinformatics(part-2)
G. N. Khalsa CollegeJul 24, 2018 1
Contents
Energy Minimization
1. Introduction
2. Potential energy surface
3. Basicsof energy minimization
Energy Minimization Algorithms
1. Steepest Descent
2. ConjugateGradient
Jul 24, 2018 2
Introduction
• energy minimization is the process of finding an arrangement
in space of a collection of atoms where, according to some
computational model of chemical bonding, the net inter-
atomic force on each atom is acceptably close to zero and the
position on the potential energy surface (PES) is a stationary
point.
single molecule, an ion,
a transition state or even a
collection of any of these
Quantum mechanics
Jul 24, 2018 3
General Terms
• A potential energy surface (PES) describes the
energy of a system, especially a collection of atoms, in
terms of certain parameters, normally the positions of
theatoms.
Explores properties of structures
Jul 24, 2018 4
Forexample
•When optimizing the geometry
of a water molecule, our aim is
to obtain thehydrogen-oxygen
bond lengths and thehydrogen-
oxygen-hydrogen bond angle
which minimizetheforcesthat
would otherwisebepulling atoms
together or pushing them apart.
Fig1: PES for water molecule: Shows the energy minimum
corresponding to optimized molecular structure for water- O-H bond
length of 0.0958nm and H-O-H bond angle of 104.5°
Jul 24, 2018 5
Significance of geometry
optimization
• Physically significant structures.
• Optimized structures often correspond to a substance as it is
found in nature.
• Optimized structurescan beused in avariety of
experimental and theoretical investigationsin thefieldsof
chemical structure, thermodynamics,chemicalkinetics.
Jul 24, 2018 6
Molecular Geometry and
Mathematical Interpretation
Thegeometry of aset of atoms 
Vector of theatoms' positions
Described byDescribed by
Single
atom
Cartesian
Coordinates
Cartesian
Coordinates
Molecule
Internal
Coordinates
Internal
Coordinates
Bond lengths , Bond
Angles , Dihedral Angles
Bond lengths , Bond
Angles , Dihedral Angles
Jul 24, 2018 7
Mathematical Interpretation
• r– A vectorthe describes the atoms' positions
• E(r) – Energy as a function of position
• Mathematically, our aim is to to find the
value of r for which E(r) is at a local
minimum.
Smallest value of a function
Obtained by calculating
∂E/∂r
Jul 24, 2018 8
Mathematical Interpretation
• Second aim is to find the curvature of the
PES at r.
Obtained by calculating
 ∂∂E/∂ri∂rj
Hessian matrixHessian matrixJul 24, 2018 9
 Thecomputational model that providesan approximate E(r) could bebased
on 
1. Quantum mechanics
2. Forcefields
3. Or acombination of thosein caseof QM/MM.
 Using thiscomputational model and an initial information of thecorrect
geometry, an iterativeoptimization procedureisfollowed, for example:
 Calculate the force on each atom (that is, -∂ E/∂ r).
 If the force is less than some threshold, finish.
 Otherwise, move the atoms by some computed step .
 Repeat from the start.
Jul 24, 2018 10
Energy Optimization
Algorithms
• An optimization algorithm can use some or all
of E(r) , ∂E/∂r and ∂∂E/∂ri∂rj to try to minimize the
forces.
Jul 24, 2018 11
Steepest Descent
Jul 24, 2018 12
Steepest Descent
Jul 24, 2018 13
Fig2 : Steepest descent
• Slow near minimum.
• Used for structures away
from minimum.
• Used asrough and
introductory method
followed by moreadvanced
methods.
Conjugate Gradient
Jul 24, 2018 14
Valuefrom
previousstep
Conjugate Gradient
Jul 24, 2018 15
Fig3: ConjugateGradient
• Used for larger system.
• Morecomputational power
and requirements.
• Moreefficient
convergence.
Jul 24, 2018 16

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Energy minimization

  • 1. Energy Minimization Aishwarya M. Rane M.Sc. Bioinformatics(part-2) G. N. Khalsa CollegeJul 24, 2018 1
  • 2. Contents Energy Minimization 1. Introduction 2. Potential energy surface 3. Basicsof energy minimization Energy Minimization Algorithms 1. Steepest Descent 2. ConjugateGradient Jul 24, 2018 2
  • 3. Introduction • energy minimization is the process of finding an arrangement in space of a collection of atoms where, according to some computational model of chemical bonding, the net inter- atomic force on each atom is acceptably close to zero and the position on the potential energy surface (PES) is a stationary point. single molecule, an ion, a transition state or even a collection of any of these Quantum mechanics Jul 24, 2018 3
  • 4. General Terms • A potential energy surface (PES) describes the energy of a system, especially a collection of atoms, in terms of certain parameters, normally the positions of theatoms. Explores properties of structures Jul 24, 2018 4
  • 5. Forexample •When optimizing the geometry of a water molecule, our aim is to obtain thehydrogen-oxygen bond lengths and thehydrogen- oxygen-hydrogen bond angle which minimizetheforcesthat would otherwisebepulling atoms together or pushing them apart. Fig1: PES for water molecule: Shows the energy minimum corresponding to optimized molecular structure for water- O-H bond length of 0.0958nm and H-O-H bond angle of 104.5° Jul 24, 2018 5
  • 6. Significance of geometry optimization • Physically significant structures. • Optimized structures often correspond to a substance as it is found in nature. • Optimized structurescan beused in avariety of experimental and theoretical investigationsin thefieldsof chemical structure, thermodynamics,chemicalkinetics. Jul 24, 2018 6
  • 7. Molecular Geometry and Mathematical Interpretation Thegeometry of aset of atoms  Vector of theatoms' positions Described byDescribed by Single atom Cartesian Coordinates Cartesian Coordinates Molecule Internal Coordinates Internal Coordinates Bond lengths , Bond Angles , Dihedral Angles Bond lengths , Bond Angles , Dihedral Angles Jul 24, 2018 7
  • 8. Mathematical Interpretation • r– A vectorthe describes the atoms' positions • E(r) – Energy as a function of position • Mathematically, our aim is to to find the value of r for which E(r) is at a local minimum. Smallest value of a function Obtained by calculating ∂E/∂r Jul 24, 2018 8
  • 9. Mathematical Interpretation • Second aim is to find the curvature of the PES at r. Obtained by calculating  ∂∂E/∂ri∂rj Hessian matrixHessian matrixJul 24, 2018 9
  • 10.  Thecomputational model that providesan approximate E(r) could bebased on  1. Quantum mechanics 2. Forcefields 3. Or acombination of thosein caseof QM/MM.  Using thiscomputational model and an initial information of thecorrect geometry, an iterativeoptimization procedureisfollowed, for example:  Calculate the force on each atom (that is, -∂ E/∂ r).  If the force is less than some threshold, finish.  Otherwise, move the atoms by some computed step .  Repeat from the start. Jul 24, 2018 10
  • 11. Energy Optimization Algorithms • An optimization algorithm can use some or all of E(r) , ∂E/∂r and ∂∂E/∂ri∂rj to try to minimize the forces. Jul 24, 2018 11
  • 13. Steepest Descent Jul 24, 2018 13 Fig2 : Steepest descent • Slow near minimum. • Used for structures away from minimum. • Used asrough and introductory method followed by moreadvanced methods.
  • 14. Conjugate Gradient Jul 24, 2018 14 Valuefrom previousstep
  • 15. Conjugate Gradient Jul 24, 2018 15 Fig3: ConjugateGradient • Used for larger system. • Morecomputational power and requirements. • Moreefficient convergence.